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G = C13×C8⋊C22order 416 = 25·13

Direct product of C13 and C8⋊C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C13×C8⋊C22, D82C26, C52.63D4, C1047C22, SD161C26, M4(2)⋊1C26, C52.48C23, C8⋊(C2×C26), C4○D42C26, D42(C2×C26), (C13×D8)⋊6C2, (C2×D4)⋊5C26, Q82(C2×C26), (D4×C26)⋊14C2, (C2×C26).24D4, C26.78(C2×D4), C4.14(D4×C13), C2.15(D4×C26), (C13×SD16)⋊5C2, C4.5(C22×C26), C22.5(D4×C13), (D4×C13)⋊11C22, (C13×M4(2))⋊5C2, (C2×C52).69C22, (Q8×C13)⋊10C22, (C13×C4○D4)⋊7C2, (C2×C4).10(C2×C26), SmallGroup(416,197)

Series: Derived Chief Lower central Upper central

C1C4 — C13×C8⋊C22
C1C2C4C52D4×C13C13×D8 — C13×C8⋊C22
C1C2C4 — C13×C8⋊C22
C1C26C2×C52 — C13×C8⋊C22

Generators and relations for C13×C8⋊C22
 G = < a,b,c,d | a13=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >

Subgroups: 116 in 68 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C8 [×2], C2×C4, C2×C4, D4, D4 [×2], D4 [×2], Q8, C23, C13, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C26, C26 [×4], C8⋊C22, C52 [×2], C52, C2×C26, C2×C26 [×5], C104 [×2], C2×C52, C2×C52, D4×C13, D4×C13 [×2], D4×C13 [×2], Q8×C13, C22×C26, C13×M4(2), C13×D8 [×2], C13×SD16 [×2], D4×C26, C13×C4○D4, C13×C8⋊C22
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C13, C2×D4, C26 [×7], C8⋊C22, C2×C26 [×7], D4×C13 [×2], C22×C26, D4×C26, C13×C8⋊C22

Smallest permutation representation of C13×C8⋊C22
On 104 points
Generators in S104
(1 2 3 4 5 6 7 8 9 10 11 12 13)(14 15 16 17 18 19 20 21 22 23 24 25 26)(27 28 29 30 31 32 33 34 35 36 37 38 39)(40 41 42 43 44 45 46 47 48 49 50 51 52)(53 54 55 56 57 58 59 60 61 62 63 64 65)(66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)
(1 60 80 102 30 16 74 47)(2 61 81 103 31 17 75 48)(3 62 82 104 32 18 76 49)(4 63 83 92 33 19 77 50)(5 64 84 93 34 20 78 51)(6 65 85 94 35 21 66 52)(7 53 86 95 36 22 67 40)(8 54 87 96 37 23 68 41)(9 55 88 97 38 24 69 42)(10 56 89 98 39 25 70 43)(11 57 90 99 27 26 71 44)(12 58 91 100 28 14 72 45)(13 59 79 101 29 15 73 46)
(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 40)(23 41)(24 42)(25 43)(26 44)(53 95)(54 96)(55 97)(56 98)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 92)(64 93)(65 94)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)
(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 53)(23 54)(24 55)(25 56)(26 57)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 92)(51 93)(52 94)

G:=sub<Sym(104)| (1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,60,80,102,30,16,74,47)(2,61,81,103,31,17,75,48)(3,62,82,104,32,18,76,49)(4,63,83,92,33,19,77,50)(5,64,84,93,34,20,78,51)(6,65,85,94,35,21,66,52)(7,53,86,95,36,22,67,40)(8,54,87,96,37,23,68,41)(9,55,88,97,38,24,69,42)(10,56,89,98,39,25,70,43)(11,57,90,99,27,26,71,44)(12,58,91,100,28,14,72,45)(13,59,79,101,29,15,73,46), (14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,40)(23,41)(24,42)(25,43)(26,44)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,92)(64,93)(65,94)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84), (14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,53)(23,54)(24,55)(25,56)(26,57)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,92)(51,93)(52,94)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,60,80,102,30,16,74,47)(2,61,81,103,31,17,75,48)(3,62,82,104,32,18,76,49)(4,63,83,92,33,19,77,50)(5,64,84,93,34,20,78,51)(6,65,85,94,35,21,66,52)(7,53,86,95,36,22,67,40)(8,54,87,96,37,23,68,41)(9,55,88,97,38,24,69,42)(10,56,89,98,39,25,70,43)(11,57,90,99,27,26,71,44)(12,58,91,100,28,14,72,45)(13,59,79,101,29,15,73,46), (14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,40)(23,41)(24,42)(25,43)(26,44)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,92)(64,93)(65,94)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84), (14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,53)(23,54)(24,55)(25,56)(26,57)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,92)(51,93)(52,94) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13),(14,15,16,17,18,19,20,21,22,23,24,25,26),(27,28,29,30,31,32,33,34,35,36,37,38,39),(40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65),(66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,60,80,102,30,16,74,47),(2,61,81,103,31,17,75,48),(3,62,82,104,32,18,76,49),(4,63,83,92,33,19,77,50),(5,64,84,93,34,20,78,51),(6,65,85,94,35,21,66,52),(7,53,86,95,36,22,67,40),(8,54,87,96,37,23,68,41),(9,55,88,97,38,24,69,42),(10,56,89,98,39,25,70,43),(11,57,90,99,27,26,71,44),(12,58,91,100,28,14,72,45),(13,59,79,101,29,15,73,46)], [(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,40),(23,41),(24,42),(25,43),(26,44),(53,95),(54,96),(55,97),(56,98),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,92),(64,93),(65,94),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84)], [(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,65),(22,53),(23,54),(24,55),(25,56),(26,57),(40,95),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,92),(51,93),(52,94)])

143 conjugacy classes

class 1 2A2B2C2D2E4A4B4C8A8B13A···13L26A···26L26M···26X26Y···26BH52A···52X52Y···52AJ104A···104X
order1222224448813···1326···2626···2626···2652···5252···52104···104
size112444224441···11···12···24···42···24···44···4

143 irreducible representations

dim111111111111222244
type+++++++++
imageC1C2C2C2C2C2C13C26C26C26C26C26D4D4D4×C13D4×C13C8⋊C22C13×C8⋊C22
kernelC13×C8⋊C22C13×M4(2)C13×D8C13×SD16D4×C26C13×C4○D4C8⋊C22M4(2)D8SD16C2×D4C4○D4C52C2×C26C4C22C13C1
# reps112211121224241212111212112

Matrix representation of C13×C8⋊C22 in GL4(𝔽313) generated by

48000
04800
00480
00048
,
0010
000312
0100
1000
,
1000
031200
000312
003120
,
1000
0100
003120
000312
G:=sub<GL(4,GF(313))| [48,0,0,0,0,48,0,0,0,0,48,0,0,0,0,48],[0,0,0,1,0,0,1,0,1,0,0,0,0,312,0,0],[1,0,0,0,0,312,0,0,0,0,0,312,0,0,312,0],[1,0,0,0,0,1,0,0,0,0,312,0,0,0,0,312] >;

C13×C8⋊C22 in GAP, Magma, Sage, TeX

C_{13}\times C_8\rtimes C_2^2
% in TeX

G:=Group("C13xC8:C2^2");
// GroupNames label

G:=SmallGroup(416,197);
// by ID

G=gap.SmallGroup(416,197);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,3818,9364,4690,88]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations

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